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Orthogonal Polynomials on the Unit Circle: Part 1: Classical Theory; Part 2: Spectral Theory
Barry Simon
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Colloquium Publications
2004; 1044 pp; softcover
Volume: 54
ISBN-10: 0-8218-4867-4
ISBN-13: 978-0-8218-4867-8
List Price: US$164
Member Price: US$131.20
Order Code: COLL/54.S
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This two-part volume gives a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrödinger operators.

Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegő's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by \(z\) (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line.

The book is suitable for graduate students and researchers interested in analysis.

Readership

Graduate students and research mathematicians interested in analysis.

Reviews

"Simon's work is not just a book about orthogonal polynomials but also about probability measures on one-dimensional Schrödinger operators and operator theory. It is extremely complex, multilayered, fascinating, and inspiring, while remaining very readable (even for advanced students). Without a doubt this monograph will become the standard reference for the theory of orthogonal polynomials on the unit circle for a long time to come."

-- Jahresbericht der DMV

"Undoubtedly that ... this book will become a standard reference in the field tracing the way for future investigations on orthogonal polynomials and their applications. Combining methods from various areas of analysis (calculus, real analysis, functional analysis, complex analysis) as well as by the importance of the orthogonal pholynomials in applications, the book will have a large audience including researchers in mathematics, physics, (and) engineering."

-- Stefan Cobzas, Studia Universitatis Babes-Bolyai, Mathematica

Table of Contents

  • Part 1: The Basics
  • Szegő's theorem
  • Tools for Geronimus' theorem
  • Matrix representations
  • Baxter's theorem
  • The strong Szegő theorem
  • Verblunsky coefficients with rapid decay
  • The density of zeros
  • Bibliography
  • Author index
  • Subject index
  • Part 2: Rakhmanov's theorem and related issues
  • Techniques of spectral analysis
  • Periodic Verblunsky coefficients
  • Spectral analysis of specific classes of Verblunsky coefficients
  • The connection to Jacobi matrices
  • Appendix A. Reader's guide: Topics and formulae
  • Appendix B. Perspectives
  • Appendix C. Twelve great papers
  • Appendix D. Conjectures and open questions
  • Bibliography
  • Author index
  • Subject index
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